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31EN  OF MATHEMATICS
offensive, few have mastered his weapons and some, unable to
bend Ms bow, insinuate that it is worthless in a practical attack.
Nevertheless Poincare is not without forceful champions whose
conquests would have been impossible to the men of the pre-
Poincare era.
Poincare's first (1889) great success in mathematical astro-
nomy grew out of an unsuccessful attack on 'the problem of n
bodies.' For n = 2 the problem was completely solved by
Newton; the famous 'problem of three bodies' (n = 3) -will be
noticed later: when n exceeds 3 some of the reductions applic-
able to the ease n = 3 can be carried over.
According to the Newtonian law of gravitation two particles
of masses m, M at a distance D apart attract one another -with
yn X -M
a force proportional to --. Imagine n material particles
distributed in any manner hi space; the masses, initial motions'
and the mutual distances of all the particles are assumed known
at a given instant. If they attract one another according to the
Newtonian law, what will be their positions and motions (velo-
cities) after any stated lapse of time? For the purposes of mathe-
matical astronomy the stars in a cluster, or in a galaxy, or in a
cluster of galaxies, may be thought of as material particles
attracting one another according to the Newtonian law* The
'problem of n bodies' thus amounts - in one of its applications -
to asking what will be the aspect of the heavens a year from
now, or a billion years hence, it being assumed that we hara
sufficient observational data to describe the general configura-
tion now. The problem of course is tremendously complicated
by radiation - the masses of the stars do not remain constant
for millions of years; but a complete, calculable solution of the
problem of n bodies in its Newtonian form would probably give
results of an accuracy sufficient for all human purposes - the
human race will likely be extinct long before radiation can
introduce observable inaccuracies.
This was substantially the problem proposed for the prize
ottered by King Oscar II of Sweden in 1887. Poincar6 did not
solve the problem, but in 1889 he was awarded the prize any-
how by a jury consisting of Weierstrass, Hermite, and Mittag-
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